Ye.A. Gayev ( ) (Institute of Hydromechanics of NASU, Kyiv, Ukraine) S.Z. Shikhaliev ( ) (Institute of Energy Problem Modelling of NASU, Kyiv, Ukraine) Viscous flow through a duct with easily penetrable roughness modelled by Navier-Stokes equations

The problem how viscous flow enters a tube or a duct, how an initially uniform velocity distribution is transformed over the initial region of the duct with smooth walls has been known in engineering and computational fluid dynamics for a long time. Schiller [1] gave an engineering solution yet in Later, in the 1950s, Schlichting [2] suggested a more sophisticated solution based on boundary layer theory and determined the length of the duct's initial region length Lx as a function of Reynolds number Re, Figure 1. In the 1970s, the problem of viscous flow entering a duct served as a keystone problem for examining first algorithms of numerical computer solutions of the complete Navier -- Stokes equations. Let now a plane semi-infinite duct with a width H=1 taken as a unit length have some easily penetrable roughnesses (EPRs, or porous layers) near the bottom and the top walls, Figure 2,A. The EPRs may be characterized by their non-dimensional heights h 1 and h 2 and densities A 1 and A 2 [3]. The problem is to predict flow transformation while entering such duct. Introduction

Engineering aspect of the presented work lies in its applicability to a range of hydraulic water supply problems with simultaneous water filtering that is evident from Figure 2. Filter porosities A 1 and A 2 and their heights h 1 and h 2 are important design parameters. Another practical example is a duct with porous filter in the centre of the duct and empty areas near walls. One more flow pattern has been shown in the Fig. 2,C. Here, the porous filter is taken of a finite length. A vortical zone may, or may not, appear behind the filters depending on the filter heights and their densities (porosities). Another motivation of the present work was to examine validity for some simplified approaches used in wind engineering and other disciplines dealing with flows over penetrable obstruction layers [3,4,5]. Let us simulate the EPR by a local force "smeared out" within the penetrable layers and and being proportional to some power k of the local flow velocity V.

Problems already considered in the fluid mechanics literature ♪ L.Schiller (1932) ♪ G.Schlichting (1934) ♪ Initial region length L x =aRe, where a=0,030 – 0,045 [1,2] ♪ First algorithms for Navier-Stokes equations were tested on this problem [1970th] ♪ G. Hagen (1839) ♪ G. Poiseuille (1840) ♪ ♪ Figure 1

Some arrangements of the EPR within the duct Figure 2 (A)

Problem 1: fully developed flow on the main duct's region has an analytical solution for governing ODEs

Problem1: Hydraulic resistance of the EPR Typical velocity and shear distributions in the fully developed duct flow with EPRs

Problem 2: viscous flow entering a duct (initial region) Non-dimensional variables Boundary conditions:

Problem 2: some results Gradual flow transformation along the duct depending on A and h Pressure in ducts with smooth walls and with EPRs

Problem 2: length of the duct's initial region L x =0,04 Re

Problem 3: flow behind finite length EPRs Рис.. Два случая течения за пористой ступенькой: внизу ; вверху () (A) (B) Vortices behind the penetrable step: (A) Dense EPR A=100, Re=100; (A) Light EPR A=10, Re=100.

Conclusion Effects of the (easily) penetrable roughness, EPR, in the duct may be accounted by a distributed force included into the hydrodynamics equations. To closure the problem, additional condition is required "at the end of initial region" that accounts for a dependence of pressure gradient β (hydraulic resistance λ) on Reynolds number along with the EPR parameters A and h to be found by consideration of 1d problem governing the "main flow region". The "closed" problem for the Navier-Stokes equations allows detailed performance of the flow field, especially on the initial flow region, what leads to a practically important relationship between its length L x and parameters of the EPR that generalizes known Schlichting's formulae [2]. In the extreme cases A→0 or h→0, the initial region length L x behaves like a Re for large Reynolds numbers but in an other way for small Re. This should be interpreted as a confirmation of the boundary layer approach for large Re. Two types of the force were considered, i.e. a linear k=1 and quadratic force k=2. The first case that allows for some analytical estimations may often provide a sufficient approach for the second one. The EPR of a finite length (penetrable backward facing step) may also be considered in the approach developed. A large vortical structures may appear behind the step, or may not, depending on the parameters A, h and Re. Reliability of the results is based upon careful consideration of a set of problems. Results may be used in hydraulics, biological fluid mechanics and as a methodological guide to some complex flows in obstructed geometries.

References 1. Schiller L. Strömung in Rohren. Leipzig, – 230 S. 2. Schlichting G. Grenzschicht-Theorie. Karlsruhe, Gayev Ye.A. Models of easily penetrable roughness for Nature and Engineering., Kiev, (to be published in Russian with extended abstract in English) 4. Finnigan J.J., Brunet Y. Turbulent airflow in forests on flat and hilly terrain. In: Wind and Trees (Edited by M.P.Coutts and J.Grace). Cambridge University Press, 1995, pp Britter R.E., Hanna S.R. Flow and Dispersion in Urban Areas. – Annual Review of Fluid Mechanics, 35, 2003, pp. 469 – Naot D., Nezu I., Nakagawa H. Hydrodynamic bahaviour of partly vegetated open channels. // J. of Hydraulic Engineering, pp , Roach P.J. Computational Fluid Dynamics. Hermosa Publishers, Albuquerque, Shikhaliev S.Z. On the Efficiency of Solving of Initial-Boundary Value Problems for Parabolic Equations by Algorithm of Polynomial Acceleration. In: 4th Intern. Conf. on Inform. System, Anal. and Synth., Vol. 2, Orlando, USA, pp